Found problems: 85335
1990 French Mathematical Olympiad, Problem 4
(a) What is the maximum area of a triangle with vertices in a given square (or on its boundary)?
(b) What is the maximum volume of a tetrahedron with vertices in a given cube (or on its boundary)?
1969 German National Olympiad, 4
Solve the system of equations:
$$|\log_2(x + y)| + | \log_2(x - y)| = 3$$
$$xy = 3$$
2016 Germany Team Selection Test, 1
The two circles $\Gamma_1$ and $\Gamma_2$ with the midpoints $O_1$ resp. $O_2$ intersect in the two distinct points $A$ and $B$. A line through $A$ meets $\Gamma_1$ in $C \neq A$ and $\Gamma_2$ in $D \neq A$. The lines $CO_1$ and $DO_2$ intersect in $X$.
Prove that the four points $O_1,O_2,B$ and $X$ are concyclic.
2024 China National Olympiad, 4
Let $a_1, a_2, \ldots, a_{2023}$ be nonnegative real numbers such that $a_1 + a_2 + \ldots + a_{2023} = 100$. Let $A = \left \{ (i,j) \mid 1 \leqslant i \leqslant j \leqslant 2023, \, a_ia_j \geqslant 1 \right\}$. Prove that $|A| \leqslant 5050$ and determine when the equality holds.
[i]Proposed by Yunhao Fu[/i]
2004 May Olympiad, 5
On a $ 9\times 9$ board, divided into $1\times 1$ squares, pieces of the form
Each piece covers exactly $3$ squares.
(a) Starting from the empty board, what is the maximum number of pieces that can be placed?
(b) Starting from the board with $3$ pieces already placed as shown in the diagram below, what is the maximum number of pieces that can be placed?
[img]https://cdn.artofproblemsolving.com/attachments/d/4/3bd010828accb2d1811d49eb17fa69662ff60d.gif[/img]
2024 Germany Team Selection Test, 1
For positive integers $n$ and $k \geq 2$, define $E_k(n)$ as the greatest exponent $r$ such that $k^r$ divides $n!$. Prove that there are infinitely many $n$ such that $E_{10}(n) > E_9(n)$ and infinitely many $m$ such that $E_{10}(m) < E_9(m)$.
2014 Purple Comet Problems, 9
Find $n$ such that\[\frac{1!\cdot2!\cdot3!\cdots10!}{(1!)^2(3!)^2(5!)^2(7!)^2(9!)^2}=15\cdot2^n.\]
2014 Iran Team Selection Test, 2
Point $D$ is an arbitary point on side $BC$ of triangle $ABC$. $I$,$I_1$ and$I_2$ are the incenters of triangles $ABC$,$ABD$ and $ACD$ respectively. $M\not=A$ and $N\not=A$ are the intersections of circumcircle of triangle $ABC$ and circumcircles of triangles $IAI_1$ and $IAI_2$ respectively. Prove that regardless of point $D$, line $MN$ goes through a fixed point.
2004 Switzerland Team Selection Test, 5
A brick has the shape of a cube of size $2$ with one corner unit cube removed. Given a cube of side $2^{n}$ divided into unit cubes from which an arbitrary unit cube is removed, show that the remaining figure can be built using the described bricks.
2002 Iran Team Selection Test, 4
$O$ is a point in triangle $ABC$. We draw perpendicular from $O$ to $BC,AC,AB$ which intersect $BC,AC,AB$ at $A_{1},B_{1},C_{1}$. Prove that $O$ is circumcenter of triangle $ABC$ iff perimeter of $ABC$ is not less than perimeter of triangles $AB_{1}C_{1},BC_{1}A_{1},CB_{1}A_{1}$.
1989 China Team Selection Test, 4
Given triangle $ABC$, squares $ABEF, BCGH, CAIJ$ are constructed externally on side $AB, BC, CA$, respectively. Let $AH \cap BJ = P_1$, $BJ \cap CF = Q_1$, $CF \cap AH = R_1$, $AG \cap CE = P_2$, $BI \cap AG = Q_2$, $CE \cap BI = R_2$. Prove that triangle $P_1 Q_1 R_1$ is congruent to triangle $P_2 Q_2 R_2$.
1997 Slovenia National Olympiad, Problem 4
The expression $*3^5*3^4*3^3*3^2*3*1$ is given. Ana and Branka alternately change the signs $*$ to $+$ or $-$ (one time each turn). Can Branka, who plays second, do this so as to obtain an expression whose value is divisible by $7$?
2025 Kosovo National Mathematical Olympiad`, P2
Find all natural numbers $n$ such that $\frac{\sqrt{n}}{2}+\frac{10}{\sqrt{n}}$ is a natural number.
1954 AMC 12/AHSME, 36
A boat has a speed of $ 15$ mph in still water. In a stream that has a current of $ 5$ mph it travels a certain distance downstream and returns. The ratio of the average speed for the round trip to the speed in still water is:
$ \textbf{(A)}\ \frac{5}{4} \qquad
\textbf{(B)}\ \frac{1}{1} \qquad
\textbf{(C)}\ \frac{8}{9} \qquad
\textbf{(D)}\ \frac{7}{8} \qquad
\textbf{(E)}\ \frac{9}{8}$
2001 IMO Shortlist, 1
Prove that there is no positive integer $n$ such that, for $k = 1,2,\ldots,9$, the leftmost digit (in decimal notation) of $(n+k)!$ equals $k$.
2019 Stars of Mathematics, 3
On a board the numbers $(n-1, n, n+1)$ are written where $n$ is positive integer. On a move choose 2 numbers $a$ and $b$, delete them and write $2a-b$ and $2b-a$. After a succession of moves, on the board there are 2 zeros. Find all possible values for $n$.
Proposed by Andrei Eckstein
2016 Mathematical Talent Reward Programme, MCQ: P 2
Let $f$ be a function satisfying $f(x+y+z)=f(x)+f(y)+f(z)$ for all integers $x$, $y$, $z$. Suppose $f(1)=1$, $f(2)=2$. Then $\lim \limits_{n\to \infty} \frac{1}{n^3} \sum \limits_{r=1}^n 4rf(3r)$ equals
[list=1]
[*] 4
[*] 6
[*] 12
[*] 24
[/list]
1950 AMC 12/AHSME, 33
The number of circular pipes with an inside diameter of $1$ inch which will carry the same amount of water as a pipe with an inside diameter of $6$ inches is:
$\textbf{(A)}\ 6\pi \qquad
\textbf{(B)}\ 6 \qquad
\textbf{(C)}\ 12 \qquad
\textbf{(D)}\ 36 \qquad
\textbf{(E)}\ 36\pi$
1997 Vietnam National Olympiad, 1
Let $ k \equal{} \sqrt[3]{3}$.
a, Find all polynomials $ p(x)$ with rationl coefficients whose degree are as least as possible such that $ p(k \plus{} k^2) \equal{} 3 \plus{} k$.
b, Does there exist a polynomial $ p(x)$ with integer coefficients satisfying $ p(k \plus{} k^2) \equal{} 3 \plus{} k$
2023 Vietnam National Olympiad, 7
Let $\triangle{ABC}$ be a scalene triangle with orthocenter $H$ and circumcenter $O$. Incircle $(I)$ of the $\triangle{ABC}$ is tangent to the sides $BC,CA,AB$ at $M,N,P$ respectively. Denote $\Omega_A$ to be the circle passing through point $A$, external tangent to $(I)$ at $A'$ and cut again $AB,AC$ at $A_b,A_c$ respectively. The circles $\Omega_B,\Omega_C$ and points $B',B_a,B_c,C',C_a,C_b$ are defined similarly.
$a)$ Prove $B_cC_b+C_aA_c+A_bB_a \ge NP+PM+MN$.
$b)$ Suppose $A',B',C'$ lie on $AM,BN,CP$ respectively. Denote $K$ as the circumcenter of the triangle formed by lines $A_bA_c,B_cB_a,C_aC_b.$ Prove $OH//IK$.
2005 Tournament of Towns, 5
In a certain big city, all the streets go in one of two perpendicular directions. During a drive in the city, a car does not pass through any place twice, and returns to the parking place along a street from which it started. If it has made 100 left turns, how many right turns must it have made?
[i](5 points)[/i]
2005 Tournament of Towns, 3
$M$ and $N$ are the midpoints of sides $BC$ and $AD$, respectively, of a square $ABCD$. $K$ is an arbitrary point on the extension of the diagonal $AC$ beyond $A$. The segment $KM$ intersects the side $AB$ at some point $L$. Prove that $\angle KNA = \angle LNA$.
[i](4 points)[/i]
2023 Austrian MO Regional Competition, 1
Let $a$, $b$ and $c$ be real numbers with $0 \le a, b, c \le 2$. Prove that
$$(a - b)(b - c)(a- c) \le 2.$$
When does equality hold?
[i](Karl Czakler)[/i]
2025 Vietnam Team Selection Test, 6
For each prime $p$ of the form $4k+3$ with $k \in \mathbb{Z}^+$, consider the polynomial $$Q(x)=px^{2p} - x^{2p-1} + p^2x^{\frac{3p+1}{2}} - px^{p+1} +2(p^2+1)x^p -px^{p-1}+ p^2 x^{\frac{p-1}{2}} -x + p.$$ Determine all ordered pairs of polynomials $f, g$ with integer coefficients such that $Q(x)=f(x)g(x)$.
2021 Korea National Olympiad, P1
Let $ABC$ be an acute triangle and $D$ be an intersection of the angle bisector of $A$ and side $BC$. Let $\Omega$ be a circle tangent to the circumcircle of triangle $ABC$ and side $BC$ at $A$ and $D$, respectively. $\Omega$ meets the sides $AB, AC$ again at $E, F$, respectively. The perpendicular line to $AD$, passing through $E, F$ meets $\Omega$ again at $G, H$, respectively. Suppose that $AE$ and $GD$ meet at $P$, $EH$ and $GF$ meet at $Q$, and $HD$ and $AF$ meet at $R$. Prove that $\dfrac{\overline{QF}}{\overline{QG}}=\dfrac{\overline{HR}}{\overline{PG}}$.